Inverse Problems for Sturm--Liouville Operators with Potentials in Sobolev Spaces: Uniform Stability

A. M. Savchuk ; A. A. Shkalikov

Funkcionalʹnyj analiz i ego priloženiâ, Tome 44 (2010) no. 4, pp. 34-53.

Résumé

Two inverse problems for the Sturm–Liouville operator $Ly=-y''+q(x)y$ on the interval $[0,\pi]$ are studied. For $\theta\ge0$, there is a mapping $F\colon W^{\theta}_2 \to l^{\theta}_B$, $F(\sigma)=\{s_k\}_1^\infty$, related to the first of these problems, where $W^\theta_2= W^{\theta}_2[0,\pi]$ is the Sobolev space, $\sigma =\int q$ is a primitive of the potential $q$, and $l^{\theta}_B$ is a specially constructed finite-dimensional extension of the weighted space $l^{\theta}_2$, where we place the regularized spectral data ${\mathbf s}=\{s_k\}_1^\infty$ in the problem of reconstruction from two spectra. The main result is uniform lower and upper bounds for $\|\sigma - \sigma_1\|_\theta$ via the $l^{\theta}_B$-norm $\|{\mathbf s}-{\mathbf s}_1\|_\theta$ of the difference of regularized spectral data. A similar result is obtained for the second inverse problem, that is, the problem of reconstructing the potential from the spectral function of the operator $L$ generated by the Dirichlet boundary conditions. The result is new even for the classical case $q\in L_2$, which corresponds to $\theta =1$.

Détail

Mots-clés : inverse Sturm–Liouville problem
Keywords: singular potentials, stability for inverse problems.

Bibliographie

[1] S. Albeverio, R. Hryniv, Ya. Mykytyuk, “Inverse spectral problems for Sturm–Liouville operators in impedance form”, J. Funct. Anal., 222 (2005), 147–177 | DOI | MR

[2] A. A. Alekseev, “Ustoichivost obratnoi zadachi Shturma–Liuvillya na konechnom intervale”, Dokl. AN SSSR, 287:1 (1986), 11–13 | MR | Zbl

[3] V. Ambarzumian, “Über eine Frage der Eigenwerttheorie”, Z. Phys, 53:9–10 (1929), 690–695 | DOI | Zbl

[4] G. Borg, “Eine Umkehrung der Sturm–Liouvilleschen Eigenwertaufgabe. Bestimmung der Differentialgleichung durch die Eigenwerte”, Acta Math., 78 (1946), 1–96 | DOI | MR | Zbl

[5] V. I. Bogachev, O. G. Smolyanov, Deistvitelnyi i funktsionalnyi analiz, RKhD, Izhevsk, 2009

[6] P. Deift, E. Trubowitz, “Inverse scattering on the line”, Comm. Pure Appl. Math., 32:2 (1979), 121–251 | DOI | MR | Zbl

[7] P. B. Dzhakov, B. S. Mityagin, “Zony neustoichivosti odnomernykh periodicheskikh operatorov Shrëdingera i Diraka”, UMN, 61:4 (2006), 77–182 | DOI | MR | Zbl

[8] P. Djakov, B. Mityagin, Fourier method for one dimensional Schrödinger operators with singular periodic potentials, arXiv: 0710.0237v1

[9] L. D. Faddeev, “Obratnaya zadacha kvantovoi teorii rasseyaniya”, UMN, 14:4 (1959), 57–119 | MR | Zbl

[10] L. D. Faddeev, “Svoistva $S$-matritsy odnomernogo uravneniya Shrëdingera”, Trudy MIAN im. V. A. Steklova, 73, 1964, 314–336 | MR | Zbl

[11] G. Freiling, V. Yurko, Inverse Sturm–Liouville Problems and Their Applications, Nova Sci. Publ. Corporation, 2001 | MR | Zbl

[12] M. G. Gasymov, B. M. Levitan, “Opredelenie differentsialnogo uravneniya po dvum spektram”, UMN, 19:2 (1964), 3–63 | MR | Zbl

[13] I. M. Gelfand, B. M. Levitan, “Ob opredelenii differentsialnogo uravneniya po ego spektralnoi funktsii”, Izv. AN SSSR, ser. matem., 15:4 (1951), 309–360 | MR | Zbl

[14] F. Gesztesy, “Inverse spectral theory as influenced by Barry Simon”, Spectral Theory and Mathematical Physics, A Festschrift in Honor of Barry Simon's 60th Birthday, Ergodic Shrödinger Operators, Singular Spectrum, Orthogonal Polynomials, and Inverse Spectral theory, Proc. Sympos. Pure Math., 76, part 2, eds. F. Gesztesy, P. Deift, C. Galvez, P. Perry, W. Schlag, Amer. Math. Soc., Providence, RI, 2007, 741–820 ; arXiv: 1002.0388v1 | DOI | MR | Zbl

[15] O. H. Hald, “The inverse Sturm-Liouville problem with symmetric potentials”, Acta Math., 141:3-4 (1978), 263–291 | DOI | MR | Zbl

[16] M. Hitrik, “Stability of the inverse problem in potential scattering on the real line”, Comm. Partial Differential Equations, 25:5-6 (2000), 925–955 | DOI | MR | Zbl

[17] H. Hochstadt, “The inverse Sturm–Liouville problem”, Comm. Pure Appl. Math., 26 (1973), 715–729 | DOI | MR | Zbl

[18] R. O. Hryniv, Ya. V. Mykytyuk, “1D Schrödinger operators with periodic singular potentials”, Methods Func. Anal. Topol., 7:4 (2001), 31–42 ; arXiv: math/0109129v1 | MR | Zbl

[19] R. O. Hryniv, Ya. V. Mykytyuk, “Inverse spectral problems for Sturm–Liouville operators with singular potentials”, Inverse Problems, 19:3 (2003), 665–684 | DOI | MR | Zbl

[20] R. O. Hryniv, Ya. V. Mykytyuk, “Inverse spectral problems for Sturm–Liouville operators with singular potentials, II. Reconstruction by two spectra”, Functional Analysis and its Applications, North-Holland Mathematical Studies, 197, eds. V. Kadets, W. Zelazko, North-Holland Publishing Co., Amsterdam, 2004, 97–114 | DOI | MR | Zbl

[21] R. O. Hryniv, Ya. V. Mykytyuk, “Transformation operators for Sturm–Liouville operators with singular potentials”, Math. Phys. Anal. Geom., 7:2 (2004), 119–149 | DOI | MR | Zbl

[22] R. O. Hryniv, Ya. V. Mykytyuk, “Eigenvalue asymptotics for Sturm–Liouville operators with singular potentials”, J. Funct. Anal., 238:1 (2006), 27–57 | DOI | MR | Zbl

[23] R. O. Hryniv, Ya. V. Mykytyuk, “Inverse spectral problems for Sturm–Liouville operators with singular potentials. IV. Potentials in the Sobolev space scale”, Proc. Edinb. Math. Soc. (2), 49:2 (2006), 309–329 | DOI | MR | Zbl

[24] E. L. Korotyaev, D. S. Chelkak, “Obratnaya zadacha Shturma–Liuvillya so smeshannymi kraevymi usloviyami”, Algebra i analiz, 21:5 (2009), 114–137 | MR | Zbl

[25] M. G. Krein, “Reshenie obratnoi zadachi Shturma–Liuvillya”, Dokl. AN SSSR, 76 (1951), 21–24 | MR

[26] M. G. Krein, “O metode effektivnogo resheniya obratnoi kraevoi zadachi”, Dokl. AN SSSR, 94 (1954), 987–990 | MR

[27] N. Levinson, “The inverse Sturm–Liouville problem”, Mat. Tidsskr. B, 1949, 25–30 | MR

[28] B. M. Levitan, “Ob opredelenii differentsialnogo uravneniya Shturma–Liuvillya po dvum spektram”, Izv. AN SSSR, ser. matem., 28:1 (1964), 63–78 | MR | Zbl

[29] B. M. Levitan, Obratnye zadachi Shturma–Liuvillya, Nauka, M., 1984 | MR

[30] J. R. Mclaughlin, “Stability theorems for two inverse spectral problems”, Inverse Problems, 4 (1988), 529–540 | DOI | MR | Zbl

[31] M. M. Malamud, Spectral analysis of Volterra operators and inverse problems for systems of ordinary differential equations, SfB Preprint No. 269, Berlin, June 1997

[32] V. A. Marchenko, “Nekotorye zadachi v teorii differentsialnogo operatora vtorogo poryadka”, Dokl. AN SSSR, 72 (1950), 457–460 | Zbl

[33] V. A. Marchenko, “Nekotorye voprosy teorii odnomernykh lineinykh differentsialnykh operatorov vtorogo poryadka”, Trudy MMO, 1, 1951, 328–420

[34] V. A. Marchenko, Operatory Shturma–Liuvillya i ikh prilozheniya, Naukova dumka, Kiev, 1977 | MR

[35] V. A. Marchenko, K. V. Maslov, “Ustoichivost zadachi vosstanovleniya operatora Shturma–Liuvillya po spektralnoi funktsii”, Matem. sb., 81:4 (1970), 525–551 | Zbl

[36] V. A. Marchenko, I. V. Ostrovskii, “Kharakterizatsiya spektra operatora Khilla”, Matem. sb., 97:4 (1975), 540–606 | MR | Zbl

[37] V. A. Marchenko, I. V. Ostrovskii, “Approksimatsiya periodicheskikh potentsialov konechnozonnymi”, Vestnik Khark. un-ta, 1980, no. 205, Prikl. matem. i mekhanika, vyp. 45, 4–40 | MR | Zbl

[38] M. Marletta, R. Weikard, “Weak stability for an inverse Sturm–Liouville problem with finite spectral data and complex potential”, Inverse Problems, 21:4 (2005), 1275–1290 | DOI | MR | Zbl

[39] A. Mizutani, “On the inverse Sturm–Liouville problem”, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 31:2 (1984), 319–350 | MR | Zbl

[40] J. Pöschel, E. Trubowitz, Inverse Spectral Theory, Acad. Press, Orlando, 1987 | MR | Zbl

[41] T. I. Ryabushko, “Ustoichivost vosstanovleniya operatora Shturma–Liuvillya po dvum spektram”, Teor. funkts., funkts. analiz i ego prilozh., 18, Kharkov, 1973, 176–185 | Zbl

[42] T. I. Ryabushko, “Otsenki normy raznosti dvukh potentsialov granichnoi zadachi Shturma–Liuvillya”, Teoriya funktsii, funkts. analiz i ego prilozh., 39, Kharkov, 1983, 114–117 | MR | Zbl

[43] A. M. Savchuk, A. A. Shkalikov, “Operatory Shturma–Liuvillya s singulyarnymi potentsialami”, Matem. zametki, 66:6 (1999), 897–912 | DOI | MR | Zbl

[44] A. M. Savchuk, A. A. Shkalikov, “Operatory Shturma–Liuvillya s potentsialami-raspredeleniyami”, Trudy MMO, 64, 2003, 159–219 | MR

[45] A. M. Savchuk, A. A. Shkalikov, “Inverse problem for Sturm–Liouville operators with distribution potentials: Reconstruction from two spectra”, Russian J. Math. Phys., 12:4 (2005), 507–514 | MR | Zbl

[46] A. M. Savchuk, A. A. Shkalikov, “O sobstvennykh znacheniyakh operatora Shturma–Liuvillya s potentsialami iz prostranstv Soboleva”, Matem. zametki, 80:6 (2006), 864–884 | DOI | MR | Zbl

[47] A. M. Savchuk, A. A. Shkalikov, “O svoistvakh otobrazhenii, svyazannykh s obratnymi zadachami Shturma–Liuvillya”, Trudy MIAN im. V. A. Steklova, 260, 2008, 227–247 | MR

[48] A. M. Savchuk, A. A. Shkalikov, Svoistva otobrazheniya, svyazannogo s vosstanovleniem operatora Shturma–Liuvillya po spektralnoi funktsii. Ravnomernaya ustoichivost v shkale sobolevskikh prostranstv, arXiv: 1010.5344v1

[49] A. N. Tikhonov, “O edinstvennosti resheniya zadachi elektroprovodimosti”, Dokl. AN SSSR, 69 (1949), 797–800 | Zbl

[50] V. A. Yurko, “Ob ustoichivosti vosstanovleniya operatora Shturma–Liuvillya”, Differentsialnye uravneniya i teoriya funktsii (Saratovskii universitet), 3 (1980), 113–124